TANK TRAILER STABILITY ANALYSIS

TANK TRAILER STABILITY ANALYSIS

C. Ma1 1 i kar junarao P.S. Fancher

Tec hn i c a l Memorandum

Prepared f o r :

The Fruehauf Corpora t ion

The Highway Sa fe t y Research I n s t i t u t e The U n i v e r s i t y o f Mich igan

November 1979

I / u+:iy:aj' i f : d , 2 Safety '.J I

I Research Institute \

FRuEHA(PF Dir/sioPb CORPORATION FRUEHAUF

J u l y 6 , 1 9 8 2

Mr. Pau l S. Fancher A s s i s t a n t Head P h y s i c a l F a c t o r s D i v i s i o n Highway S a f e t y Research I n s t i t u t e Huron Parkway and Bax te r Road Ann Arbor , Michigan 48109

Dear Mr. Fancher :

T h i s l e t t e r i s i n r e p l y t o your r e q u e s t of June 30, 1 9 8 2 t o p l a c e a copy of T e c h n i c a l Memorandum "Tank T r a i l e r S t a b i l i t y A n a l y s i s " i n t h e HSRI L i b r a r y .

We a g r e e t h a t t h i s memorandum should be made a v a i l a b l e t o t h e p u b l i c th rough HSRI's normal c h a n n e l s .

Yours t r u l y ,

A . F. Hu lve r son , Vice P r e s i d e n t - E n g i n e e r i n g

10900 Harper Avenue, D e t r o i t , Michigan 48213 -3389 (313) 267-1000 P. 0 . BOX 238, D e t r o i t , M I 48232 -9961

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . 1 . INTRODUCTION 1

. . . . . . . . . . . . . . . . . . . . . . . . . . 2 . SUMMARY 2

. . . . . . . . . . . . . . . . . 2 . I Directional Stabi 1 i ty 2 . . . . . . . . . . . . . . . . . . . 2.2 Rollover Thresholds 3

. . . . . . . . . . . . . . . . . . . . 3 . VEHICLE DESCRIPTION 5

. . . . . . . . . . . . . . . . . . . . 4 . DIRECTIONAL BEHAVIOR 9

. . . . . . . . . . . . 4.1 The Directional Response Model 9 . . . . . . . . . . . . . . . . . . . . . . 4.2 Eigenvalues 10

. . . . . . . . . . . . . . . . . . 4.3 Frequency Response 14 . . . . . 4.4 Transient Response During Emergency Maneuvers 18

5 . ROLL RESPONSE . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . . . . . . 5.1 Roll Model 24 . . . . . . . . . . . . 5.2 Time Histories of Lateral Force 26

. . . . . . . . . . . . . . . . . . 5.3 Simulation Results 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES 38

APPENDIX A . Parameters for Directional Response Calculations . . . . . . . . . . . . . . . . . . . . 39

APPENDIX B . Eigenvalues a t 50 mph . . . . . . . . . . . . . . . . 48

. . . . . APPENDIX C Lateral Acceleration Frequency Response Plots 51

APPENDIX D . Lateral Acceleration Time Histories During 2-Second Emergency Lane-Change Maneuvers a t 50mph . . . . . . . . . . . . . . . . . . . . . . 90

APPENDIX E . Suspension Models and Listing of Computer Program . . . . . . . . . . . . . . . . . . 129

. APPENDIX F Roll Parameters . . . . . . . . . . . . . . . . . . . 145

1 .0 INTRODUCTION

This report presents results of directional s t ab i l i t y and rollover threshold calculations for various articulated vehicle configurations incorporating tank t r a i l e r s manufactured by the Fruehauf Corporation. The vehicl es examined include three tractor-semi t ra i 1 e rs , one truck/full - t r a i l e r , and two double tankers.

The directional stabi 1 i ty and rol lover analyses were performed by the Highway Safety Research Inst i tute (HSRI) of The University of Michigan. To i n i t i a t e the study, parameters describing the t r a i l e r s and their suspension (or spring) characteristics were derived from drawings and

measurements supplied by Mr. Stan Sadlocha of the Fruehauf Corporation. Typi- cal t ractor and t i r e parameters from previous HSRI research studies were used t o complete the descriptions of the vehicles analyzed. The analysis employed computerized vehicle models original ly developed in a previous HSRI research program entit led "Ad Hoc Study of Certain Safety-Related Aspects of Doubl e-Bottom Tankers" [I].

This analysis was proposed t o Fruehauf by HSRI i n response to a l e t t e r from Mr. Larry Botkin of Fruehauf t o Mr. R.D. Ervin of HSRI. Sub- sequently, the HSRI proposal was revised according t o communications with the Research and Development Division of Fruehauf. Once the proposal was accepted, Messrs. John Getz and Stan Sadlocha of the R & D Division pro- vided liaison between HSRI and the Fruehauf Corporation.

A concise summary of the findings of th is study i s presented in the next section. Section 3 provides a technical description of the vehicles and loading configurations examined. Sections 4 and 5 describe the methods used and the results obtained i n the directional and roll analyses, respectively. Several appendices ( A through F) provide detai 1 ed information concerning ( 1 ) the meanings and val ues of the parameters needed in the computerized models, ( 2 ) the resu l t s of pertinent calculations, and ( 3 ) the new suspension (spring) models implemented in th is study.

2.0 SUMMARY

The findings of the directional stabi 1 i ty and rol lover threshold

calculations for various tank trailer configurations are summarized in

this section.

2.1 Directional Stability

The directional performance and stabi 1 i ty were evaluated using the 1 inear yaw plane model described in Reference [I]. Each vehicle was

analyzed in the fully loaded, empty and partial loading conditions. The

directional performance measures itemized below were used as the basis for quantifying the directional stabi 1 i ty of each vehicle configuration.

1) Natural frequencies and damping ratios of the natural

modes of yaw motion at 50 mph (eigenvalues).

2) Lateral acceleration frequency response of the vehicle at 50 mph.

3) The transient response exhibited by the vehicle during

a two-second, emergency-type lane-change maneuver executed at 50 mph. (This maneuver was found to be very close to a worst case situation for these vehicles.)

Numerics based on performance measures (2) and (3) were found to be

the most useful in characterizing the directional behavior of the vehicles, especial ly the tendency of the rear-most trail ers of mu1 ti -articulated

vehicles to exhibit an amp1 ified and weakly damped directional response. The following results were obtained from the directional stability calculations.

1 ) A1 1 of the three tractor-semi trailer combinations that

were analyzed exhibited we1 1 -damped and attenuated semitrailer motions.

2 ) Vehicles with typical, unmodified dollies (i .e., the full trailer of the truck/full-trailer, and the pup trailers of the conventional double tanker configurations) were found to exhibit lightly damped, amplified lateral motions in an

emergency lane-change maneuver at 50 rnph. The amp1 ification in the lateral acceleration response was found to range from

1.5 t o 3.0 in the emergency lane-change maneuver.

3 ) Rigidizing the pintle hook connection in yaw and roll was

found t o increase the damping and decrease the ampl ifying

tendency in the double tanker configurations. The ampl i -

fication in an emergency lane change was reduced by 21%

for the five-axle double tanker and by 30% for the nine-

axle double tanker.

2 . 2 Rollover Thresholds

Calculations were performed for each vehicle in the ful ly loaded

condition. A modified version of the nonlinear roll model described in

Reference [ I ] was used for computing the rollover thresholds for steady

turning and emergency lane-change-type maneuvers.

The results obtained from the roll over threshold calculations were

as follows:

1 ) The steady turning rollover threshold was found to range

from 0.37 g t o 0.46 g for a l l of the vehicles analyzed.

2 ) Vehicles equipped with Fruehauf T-type a i r suspensions

exhibited higher rollover thresholds as compared t o those

equipped with conventional leaf springs because ( a ) the

a i r springs do no t have free-play and ( b ) the a i r suspen-

sions have roll st iffnesses which are roughly comparable

t o the roll st iffnesses of the leaf spring suspensions.

3) The roll over thresh01 d during emergency maneuvers (dynamic

rol lover threshold) was found t o vary over a very wide

range. The tractor-semi t r a i 1 ers and the double tankers

equipped with rigidized pintle hooks were found t o possess

a dynamic rollover threshold which was higher (in the range

of 0.46 t o 0.64 g j than the rollover threshold in a steady

turn. The conventional double tanker and the truck-full

t r a i l e r configuration, on the other hand, exhibited a much

lower dynamic rollover threshold ( in the range of 0.18 t o 0.28 g ) .

4) Rigidizing the pintle hook of a double tanker resulted, approximately, in a twofold increase in the dynamic roll over thres hol d.

5 ) The presence of backlash in the suspensions was found

t o lower the rollover threshold during steady turning and emergency maneuvers. (The infl uence of suspension backlash on the rollover threshold of the pup t r a i l e r of

a double tanker i s i l lustrated in Figure 18,)

The directional response and rollover results for the vehicles studied i n this program are presented in summary form in two bar charts, specif ical ly , Figure 11, ent i t led "Lateral Acceleration Gain During Emergency Maneuvers," and Figure 17, ent i t led "Static and Dynamic Roll- over Thresholds." The relat ive performance of the vehicles studied may be compared by examining Figures 11 and 17.

3.0 VEHICLE DESCRIPTION

Directional and roll response calculations were performed for s ix

vehicles. The vehicles included three tractor-semi t r a i l e r s , a truck/ f u l l - t r a i l e r and two double tanker-type vehicles. Tab1 e 1 presents

schematic diagrams of the vehicles, along with other relevant information.

For each vehicle, directional response calculations were performed

in the fu l ly loaded, empty and partial loading conditions, making a total

of 38 cases, which are defined in Table 2. I t should be noted t h a t the

influence of roll on directional behavior i s neglected in the yaw plane

analysis. Hence, configurations with identical layout, b u t different

suspension properties (e.g., Ia with Ib, I I a with IIb, I Ic with IId,

and VIa w i t h VIb) are lumped together for the purposes of evaluating the i r

directional behavior.

Calculations of rollover thresholds in steady turns and rapid lane-

change maneuvers were performed for the 13 configurations ( Ia through VIc)

l is ted in Table 1 . Vehicle parameters corresponding t o a fu l ly loaded

condition were used in the roll analysis.

Data Set # and Case #

Table 2. Loading Conditions for Which Directional Response Calculations Were Performed.

Configuration

#Ia and I b - Tractor-Semi BKY8499 Ful ly loaded '

Compartment #I full Compartment #4 full

#IIa and I I b - Tractor-Semi BKD0065 -

Ful ly loaded Compartment #I full Compartment #4 full

#IIc and IId - Tractor-Semi BKD0065 Empty Compartment #I full Compartment 84 full

#I1 I - Tractor-Semi BKY940-1 Fully loaded

Empty Compartment #l full Compartment #4 full

#IV - Truck/Full-Trailer BLY2714 Fully loaded

Empty Semi loaded, pup empty Semi empty, pup loaded

#Va - 5-Axle Double Tanker BLY2985 Fully loaded

Empty Semi loaded, pup empty Semi empty, pup loaded

Table 2. (Cont.)

Data S e t # and Case # Conf igura t ion

#Vb - Modified 5-Axle Double SLY2985

F u l l y 1 oaded

E ~ P t y Semi loaded, pup empty

Semi empty, pup loaded

#VIa and VIb - 9-Axle Double BKD0067

F u l l y loaded

E ~ P t y Semi loaded, pup empty


Semi 1 oaded, pup cornp. #3 f u l l

Semi loaded , pup cornp. #2 & 3 f u l l

#VIc - Modified 9-Axle Double BKD0067

F u l l y loaded

Empty Semi loaded , pup empty


Semi 1 oaded, pup cornp. #3 f u l l

Semi loaded , pup cornp. #2 & 3 f u l l

4.0 DIRECTIONAL BEHAVIOR

A 1 inear yaw plane model ( a model in which a l l motions of the

vehicle a re restricted t o the horizontal plane) was used for examining

the directional performance of the vehicles. Numerics based on the direc-

tional performance measures itemized below have been used for quantifying

the directional s tab i l i ty and performance of the vehicles in each of the

loading conditions 1 isted in Table 2:

1 ) natural frequencies and damping ratios of the natural

modes of yaw motion (eigenvalues) a t 50 mph,

2 ) lateral acceleration frequency response of the vehicle

(by frequency response, we mean the directional response

of the vehicle t o sinusoidal s teer inputs a t 50 rnph), and

3 ) the transient response exhibited by the vehicle during

a 2-second emergency-type lane-change maneuver executed

a t 50 rnph.

4.1 The Directional Response Model

The mathematical model used in th i s study i s the same as the one

developed by HSRI for the double-bottom tanker study [ I ] in 1978. The

important simp1 ifying assumptions made in the process of deriving the

equations of motion are as follows:

1 ) Cornering forces and aligning moments generated a t the

tire-road interface are l inear functions of the s idesl ip

angle.

2 ) Pitch and roll motions of the sprung mass may be neglected.

3 ) There are no significant t i r e forces present in the longi-

tudinal direction, and the vehicle i s assumed t o have a constant forward velocity.

4) Articulation angles are small such t h a t the approximations

sin r = r and cos r = 1.0 hold.

5 ) Steering system dynamics are l e f t o u t of the model and the

steering input i s assumed t o be given direct ly t o the

front wheels.

6) The liquid in the tanks i s assumed to take part in the yawing motion w i t h o u t sloshing.

Figure 1 shows the representation of a double tanker in the yaw model. The vehicle parameters used t o describe the 38 loading conditions defined i n Table 2 are 1 isted in Appendix A. All major dimensions and weight distributions were obtained using drawings supplied by the Fruehauf Corporation, while yaw moments of iner t ia were estimated based on the s ize and mass of each u n i t . Tire characteristics corresponding to a Fruehauf 10 x 20 r ib t i r e were used for a l l vehicles expect BKD0067, where data for an 11 x 22 .5 Uniroyal Fleetmaster t i r e were used.

The complete se t of 1 inear different ial equations which describe the directional motion of the vehicle i s given in Appendix A of Reference [ I ] . These equations, when written in matrix notation, are of the form

where [A], [B] and (C) are matrices whose elements are functions of the vehicle parameters, 6 i s the front-wheel angle, and { x ) i s the vector of s t a t e variables. The s t a t e variables for a tractor-semitrai l e r , fo r example, a re

v1 - la teral velocity of t rac tor

r1 - yaw ra te of t ractor

r2 - yaw ra te of t r a i l e r

r - art iculat ion angle

Eigenval ues , frequency response functions, and time hi s tor ies of vehicle behavior during transient maneuvers were evaluated by the appl i - cation of suitable numerical algorithms t o t h i s s e t of equations. Flow diagrams for calculations of eigenvalues and transient responses are given in Appendix A of Reference [I] . The method adopted for the calula- tion of frequency response functions will be discussed in Section 4.3.

4.2 Eigenval ues

The number o f natural modes of yaw motion exhibited by an ar t iculated vehicle depends upon the number of independent ar t iculated units consti - tuting the vehicle configuration. For example, a tractor-semitrail e r

(composed of two units) has two modes of motion and a corresponding set

of two pairs of complex eigenvalues, while a conventional double tanker (consisting of a tractor, semitrailer, dolly and pup trailer) has a set of four pairs of eigenvalues. For a vehicle to be directionally stable,

it is necessary that all the natural modes of yaw motion be positively damped (that is, for stability, the damping ratio, 5 , must be greater than zero). Small values of damping ratio indicate that transient motions generated during a maneuver will consist of slowly decaying oscillations occurring at frequencies close to the natural frequencies of the lightly damped modes of motion. The natural frequencies and damping ratios of the

eigenvalues are primarily influenced by vehicle design, payload distribution, and forward speed. Appendix B contains a tabulation of the natural frequencies and damping ratios for a forward speed of 50 mph for each of the cases listed in Table 2.

The influence of payload distribution on the damping ratio of the least damped mode of each vehicle is summarized in Figure 2. Improvements in the damping of a double tanker when fitted with a rigidized pintle

hook are evident from this figure. With one exception, the tractor- semitrailer vehicles, as a class, exhibit much higher damping levels than

the tanker Full ;trailer and double tanker-type vehicles. The damping ratio of 0.51 exhibited by tractor-semi trai ler BKD0065 with compartment #3 loaded (see Fig. 2 and Case 7 in Appendix B) results because the c.g. of the semitrailer, in this loading condition, is shifted rearward to a point where it almost lies on top of the mid-axle of the three-axle suspension.

For simple dynamical systems such as a single degree of freedom spring mass damper system, the eigenvalues furnish us with enough information about the response of the system to external forcing functions, but for multiple degree of freedom systems, such as a tractor-semitrailer or a double tanker, the information gained from the eigenvalues is in-

sufficient to predict their directional behavior. Eigenval ues do not, for example, reveal the problem of rearward amp1 if ication of directional

response that is pecul iar to mu1 ti -articulated vehicles. A frequency response analysis or an analysis of the transient response of the vehicle during emergency maneuvers such as a lane change, on the other hand, provides much more information. (These methods of analysis, of course, involve a greater amount of computation than an eigenval ue analysis. )

BKD0065 Tractor - Sernietrailer

- Truck \Full-Trailer

J

0067 (1 Modified BKD 11 Double Tanker

Figure 2 . The range o f variation, with load, of the damping ra t io of the l eas t damped mode for each vehicle. Full;/ loaded condition i s marked with the symbol x.

Results obtained from the frequency response and t ransient response analyses are presented in the next two sections.

4.3 Frequency Response

The frequency response of a multi-articulated vehicle can be obtained from the different ial equations of motion (1) as follows:

A sinusoidal s teer i n p u t of unit amplitude and frequency u can be written as -

- -.

Since the system of equations are l inear , the response of the vehicle i s also harmonic and i s of the form

where {x} i s the vector of complex quantities which define the magnitude and phase of each s t a t e variable. Therefore, upon substi tution of ( 2 )

and (3) into Equation (1) and rearranging, we get

[ [ A l i ~ - [B]] {XI = CCI

and

The right-hand side of ( 5 ) can be evaluated for any given input frequency, U. - . . -.. -- ..- . - - -

The lateral acceleration response of each unit of an ar t iculated vehicle i s related t o the s t a t e variables by the following relationships:

Tractor

Semi

P u p Trailer a a = v l + u r l - ~ 1 ~ ~ 1 +x )k2 Y4 - ( ' 2 ~ 28

+x );. - ('38 3C 3 - ' 4 ~ ~ 4

where a ' s are the la teral accelerations

Y u i s the forward velocity

X 1 ~ ' X 2 ~ ' etc. are dimensions defined in Appendix A.

Hence, the frequency response of the lateral acceleration of each element of an articulated vehicle can be obtained by combining ( 6 ) with (5 ) .

Examples of the lateral acceleration frequency response of a tractor- semitrailer (BKD0065), truck/full - t r a i l e r (BLY2719) and a double tanker (BLY2985) in the baseline and modified conditions are shown in Figures 3,

4, and 5, respectively. The ordinate in these plots i s the amplitude of the 1 ateral acceleration ( f t / sec2) response in decibels for a front-wheel input amplitude of one degree. (Note: a quantity x when expressed in the decibel scale i s 20 loglO(x).) A complete se t of plots for each of the 38 cases l is ted in Table 2 are included in Appendix C.

Figure 3, which i s representative of the response exhibited by most commercial tractor-semitrailer configurations, indicates an attenuated semitrailer response (by attenuation we mean a t r a i l e r response which i s smaller than the tractor response) for frequencies greater than 2 rad/sec. Certain unfavorable loading conditions in which the rearmost compartment of the semitrailer i s loaded, such as Case #7 in Table 2 , result in a semitrailer response which i s only s l igh t ly larger than that of the tractor, in the 1 t o 2 rad/sec range. (See the results for Data Set #7

in Appendix C . ) The ful l t r a i l e r of a t ruck/ful l - t ra i ler combination (Fig. 4) or the pup t r a i l e r of a conventional double tanker (Fig. 5a) , on the contrary, exhibit considerable amp1 if icat ion of the lateral acceleration response in the 1 t o 6 rad/sec frequency range. Comparison of Figure 5a with Figure 5b reveals the attenuating influence of the rigidized pintle hook on the la teral motion of the p u p t r a i l e r .

The maximum amp1 i f ication exhibited by the rearmost t r a i l e r , over the entire frequency range, serves as a convenient measure of the directional performance of mu1 t i -ar t iculated vehicles. As shown in Figure 4 ,

the maximum amplification can be computed in the decibel scale simply by finding the maximum difference, in d b , between the frequency response of the rearmost t r a i l e r and the tractor.

TRRCTOR-SEM I BKD-0965 FULLY UIRDED ORTASET#5 Figure 3

TRUCK/ FULL -TRR I LER BLY-27 14 FULLY LORDED . DflTFISET* 15

Figure 4

DOUBLE TANKER B-7-2985 FULLY LORDED . DATASETa 19

Figure 50

MODIFIED DOUBLE BLY-2985 FULLY LOflDED .DflTflSET*23

Figure 5 b ,,

The peak ampl i f i c a t i on fac to r of each of the 38 cases analyzed a r e

p lot ted in Figure 6 in a histogram format, using the decibel scal'e. A

sca le containing the actual peak gain i s a l so superimposed on the x axis of t h i s diagram.

4.4 Transient Response During Emergency Maneuvers

The behavior of a r t i cu la ted vehicles during emergency maneuvers a t normal highway speeds serves as a good indicator of t h e i r d i rect ional and ro l l s t ab i 1 i ty. An emergency-type lane-change maneuver, f o r example, causes t he rea r t r a i l e r s of a mu1 t i -a r t i cu la ted vehicle t o experience higher l eve l s of l a t e r a l accelerat ion than the t r a c t o r , thereby making the rear t r a i l e r s more susceptible t o a ro l lover . This type of amplify- ing behavior i s primarily influenced by vehicle design and operating parameters such as speed and payload d i s t r ibu t ion .

A s t e e r input of the form shown in Figure 7 was used f o r the purpose of examining the directional response of various vehicles in a lane-change (o r obstacle avoidance) type of maneuver a t a forward speed of 50 mph.

For example, as can be seen i n Figure 8, the semi t ra i l e r of a t r a c to r - semi t r a i l e r combination does not exhibi t an ampl i f i c a t i on of the peak 1 a te ra l accelerat ion experienced by the t r a c t o r in the simulated 1 ane-change maneuver. The p u p t r a i l e r s of the t r u c k / f u l l - t r a i l e r (Fig. 9 ) and the double tanker ( F i g . IOa), on the contrary, exh ib i t considerable amplifica- t ion of the l a t e r a l accelerat ion response. Comparison of Figure 10b with Figure 10a shows the reduction of the peak p u p t r a i l e r l a t e r a l accelerat ion produced by r ig idiz ing the p in t l e hook of the double tanker.

Figure 7

18

PI 4 . Cornp # 4 FUII

Q

Y

I 5 - Fully Loaded

I . Fully Looded

2 Empty

3 . Cornp. tl I FUII

. Comp. # I Full

10. Cornp. # 4 Full

Ul (D

0

0

31 1 1 . Fully Loaded

6 . Cornp. *I Full

7 , Cornp. 34 Full

8 . Empty

Peak Galn I

-

$

Rearmost Troller Lateral Acceleration

1.2 . Empty

13. Cornp # I Full

- Frequency

14. Comp *4 Full

el Is Fully Looded

Empty

S e m Loaded, Pup Empty

18 Semi Empty , Pup Loaded

1 l9 Fully Locded

LO Q * N

lj CP

1.0 PEAK GAlN

1.2 I? 1:6 1:8 20 2,2 29 2:s 2,8 3p 3,4 3:3 4.,2 I 1 I I I I

2 0 Empty

21 BASELINE

Semi Loaded , Pup Empty

22 Seml Empty, Pup Loaded

23 Fully Loaded

24 - Empty

25 Sern~ Loaded, Pup Empty MODIFIED

26 - Sernl Empty, Rip Loaded

b

g O 0 Y

0 2.0 4 .O 6 .O 8 . 0 I 0 .O 12.0

PEAK GAlN (DB)

27 Fully Loaded

28 E m ~ t Y

29 Sem~ Lsoded, Pup Empty

30 $ern1 Empty, PUP Lcuded BASELINE

31 Sen1 Full. Puo Cornc * 3 Full

32 Sernt Full. puo 3 2 B 43 C m o Full

33 Fully Loaded

3 4 Empty

3 5 Semi Loadea, Pup Erncty MODIFIED

3 6 Sem~ Empty, Pup Loaded

37 Sern~ Full, Pup #3 Corn?. Full

38 Seml Full, Pup t 2 & 9 3 Cornp. FUU

PEAK GAINS OF LATERAL ACCELERATION RESPONSE

Figure 6 19

0 T m m e TFWILER

Tim: 050 LW 1s 200 2.50 3a0 3s a TIME (SEC)

TRflCTOR-SEMI BKD-0065 LOAOED , DATASET85

Figure 8

Fiaure 9 -

DOUBLE-TRNKER-BLY -2985, FULLY LORDED DRTASETa 19

Figure 100

MODIFI ED-DOUBLE-BLY-2985, FULLY LOADED DATflSfTr23

Figure IOb rn 7

The r a t i o between the peak l a te ra l acceleration of the rearmost

t r a i l e r and tha t of the t r ac to r serves as a good index of directional s t a b i l i t y and a lso of the extent t o which the rear t r a i l e r s a re susceptible t o a ro l lover during emergency maneuvers. Figure 11 presents a summary of the peak l a t e r a l acceleration ra t ios (gains) fo r the 38 cases analyzed in t h i s study. An inspection of Figure 11 shows t ha t a l l the three semi- t r a i l e r s (except Case #7) have peak l a te ra l accelerat ion gains which a r e

l e ss than 1.0. Both the t r uek / fu l l - t r a i l e r (BLY2714) and the base1 ine five-ax1 e double tanker (BLY2985) exhibit amp1 i f i ca t ion levels which 1 i e in the range of 1.5 t o 2.0.

Modification of the pint1 e hook reduced the average peak l a t e r a l accelerat ion gain of the five-axle double tanker (BLY2985) by 21 percent-from 1.77 t o 1.39. (By average we mean the average fo r a l l the four loading conditions.)

A 1 arger reduction of 30 percent (from 2.82 to 1.83) was produced by r ig idiz ing the p in t l e hook of the nine-axle double tanker (BKD0067).

a Tractor I F I R S T i A T i O = % SECOND RATIO : %

b2

MODIFIED

\ M O D I F I E D

t 0 I 2 3

R A T I O OF P E A K L A T E R A L A C C E L E R A T I O N

( F I R S T OR SECOND RATIO, WHICHEVER IS HIGHER)

L A T E R A L A C C E L E R A T I O N GAIN DURING EMERGENCY MANUEVERS

Figure I I

5.0 ROLL RESPONSE

The methodology used, and the results obtained, from rollover threshold calculations are presented in this chapter. A dynamic ro l l

model was used in simulating the rol l behavior of the vehicles during ( 1 ) steady turns and ( 2 ) two-second emergency-type lane-change maneuvers. Table 1 defines the thirteen vehicle configurations for which calculations were performed. Results from an investigation of the influence of sus-

pension backlash on the rollover threshold are also included in th i s chapter.

5.1 Roll Model

The rol l model used in these calculations was a modified version of

the dynamic rol l model which was developed for the Michigan double tanker study [ I ] . All motions of the vehicle are restr ic ted to the rol l plane in th is five-degree-of-freedom nonlinear rol l model. The representation of the vehicle in the roll model i s shown in Figure 12. The five degrees of freedom permitted in the model are:

1 ) la teral displacement of the unsprung mass c.g.

2 ) vertical displacement of the unsprung mass c.g.

3 ) roll of the unsprung mass with respect t o the ground

4) roll of the sprung mass with respect t o the unsprung mass

5 ) vertical motion of the sprung mass with respect to the unsprung mass.

Itemized below are the important simp1 ifying assumptions made in the process of modeling the vehicle.

1 ) All sprung and unsprung mass character is t ics which are dis-

tributed along the length of the vehicle are lumped together and are assumed t o be present in a single roll plane.

Mu1 t i -a r t icu l ated vehicles in which a1 1 units are r igidly coupled in roll (such as tractor-semi t r a i l e r s and double tankers with a rigidized pintle hook) are represented in the model by combining a l l sprung mass, unsprung mass, a n d suspension character is t ics . I n the case of vehicles

REPRESENTATION OF THE VEHICLE IN THE DYNAMIC ROLL MODEL

Figure 12

2 5

such as a conventional double tanker, where very l i t t l e ro l l coupling

exis t s between the pup t r a i l e r and the res t of the vehicle, the pup t r a i l e r alone i s represented and analyzed.

2 ) The time history of the la teral force a t the tire-road inter-

face i s assumed to be a known quantity and i s used as input for simulating the rol l response of the vehicle (see Section 3 . 2 ) .

3 ) Vertical s t i f fness and damping of the t i r e s are represented

by 1 i near springs and viscous dampers, as shown in Figure 12.

4 ) The sprung mass i s assumed to rol l about a rol l axis which i s

a t a fixed height beneath the sprung mass c.g., permitting both vertical

and rol l motion of the sprung mass with respect t o the unsprung mass.

5 ) Two suspension spring models are available as options:

a ) a suspension represented by linear springs and coulomb f r ic t ion elements, with a dead zone (which i s used to represent suspension backlash), and

b) a suspension spring model which can be used t o f i t measured force-defl ection characteristics.

The complete se t of different ial equations which describe the ro l l dynamics are presented in Reference [ I ] . A l i s t i ng of the computer program along with a description of the spring models i s given i n Appendix E.

Roll parameters of a l l the vehicle configurations along with spring data are l i s ted in Appendix F. Vehicle parameters such as sprung and unsprung masses and c.g. height were estimated from drawings supplied by the Fruehauf Corporation. The roll moments of iner t ia were estimated based on the s ize and weight of the sprung and unsprung masses. The suspensions were represented by parameters which gave a best f i t to the spring data suppl ied by the Fruehauf Corporation.

5.2 Time Histories of Lateral Force

Time histories of the la teral force a t the tire-road interface were used as input for the simulation of ro l l response. The rollover threshold for each vehicle was computed by conducting a ser ies of simulations. The la teral force level was increased in small s teps, until the rollover l imit was reached. The c r i t i ca l force level needed t o rol l the vehicle over was then used t o compute the rollover threshold in g ' s .

A forcing function of the form shown in Figure 13 was used f o r

simulating steady turns. The smooth, b u t rapidly r i s i ng , shape of the curve was chosen so as t o keep the ro l l t ransients small while a t the same time avoiding long simulation times.

Since the shape and magnitude of the l a te ra l force time history i s not known for a lane-change maneuver, i t was obtained by an indirect method which i s described below:

F i r s t the directional response of the vehicle during a two-second lane-change maneuver was simulated using the 1 inear yaw model described i n Section 4.1. I t was not possible to d i rect ly use the l a t e r a l force time history (from the directional response calcula t ion) as i n p u t fo r the ro l l simulation due t o the f a c t t ha t i t s shape differed considerably from the nonlinear response observed dur ing experiments of emergency lane-change maneuvers a t the rollover 1 irni t. (See Reference [ I ] . ) A t the rol lover 1 irni t , i t was found tha t the l a t e r a l acceleration response (and hence the l a te ra l force) was of the shape shown in Figure 14a. The vehicles, moreover, exhibited a tendency t o rol lover during the second peak in the l a te ra l acceleration time history. Hence, i n order t o make a more r e a l i s t i c estimate of the rollover thresh01 d , the l a t e r a l force time history obtained from the l inear model was modified. As shown in Figure 14c, a dwell (o r f l a t top) of 0.4 sec. was added t o the second peak of the l a t e r a l force time history obtained from the l inear model.

5.3 Simulation Results

Results of rol lover threshold computations a re presented in t h i s section. F i r s t , examples of ro l l response d u r i n g simulations of steady turning and lane-change-type maneuvers a r e shown. Then, a summary of the rollover threshold 1 eve1 s i s presented fo r a1 1 the vehicle configurations analyzed.

5.3.1 Roll Response During Simulation of Steady Turn. A l a t e r a l force time history which r i s e s smoothly to the steady-state level was used for simulating steady turns. The force inputs and the ro l l responses shown in Figures 15a and 15b a r e f o r the modified nine-axle double (BKD0067).

Time (sec) -

FORCE 0.0 600 685 750 810 855 890 920 945 965 978 986 994 1000 1000

L A T E R A L FORCE TIME HISTORY USED FOR SIMULATING STEADY T U R N S

Figure 13

Lateral Acceleration Time History Observed During Experiment

Figure 140

I

Lateral ~ o r c e ' Time History Obtained From Lineor Modal I

I I I Figure 14b

-

TIME (SEC)

MODIFIED 9AXLE DOUBLE , BKD 0067 , CONFIG *6C , STEADY TURN

Figure 150

M0DIF.IED 9AXLE DOUBLE , BKD 0067 , CONFIG +6C , STEFlDY TURN

Figure 15b - -

As can be seen in Figure 15b, the vehicle rolled over when a lateral

force level of 41,200 1 bs was applied. This lateral force of 41,200 1 bs,

when translated t o g units, gives a rollover threshold of 0.39 g 's .

Despite the use of a smoothly increasing la teral force input, some

oscil latory roll transients in the sprung mass motion (see Fig. 15b) were

found t o occur. The small errors in rollover threshold calculations, which

resul t from such an oscillatory roll motion, resul t in rollover threshold

estimates which are on the conservative side. (By conservative, we mean

t h a t the estimated rollover threshold wil l , i f a t a l l , be s l ight ly lower

than the actual threshold level during an ideal steady turn.)

5.3.2 Roll Response During Simulation of Lane-Change Maneuvers. The

lateral force inputs and the simulated roll responses for two-second

lane-change maneuvers of increasing levels of severity are shown in

Figures 1Ga and 16b. In i t ia l ly , the time history of the lateral force

input was obtained from the directional response simulation for a steering

input of one degree amplitude a t the front wheels. The severity of the

maneuver i s then indicated by the amount by which the la teral force time

hi story obtained from the directional response simulation i s ampl ified.

Typical ampl ification factors are labeled "AMP" in the lower left-hand

corner of these figures. Figure 16b indicates t h a t the roll response i s

highly nonlinear, especially a t maneuver levels which approach a rollover.

The force generated by the suspension spring model during the lane-

change maneuver i s shown in Figure 16c. As can be seen in th i s figure,

the spring goes through a cycle of tension and c~onpression during the

maneuver. A comparison of Figure 16c with Figure F.1 reveals the accuracy

with which the spring model can be made t o f i t measured spring data. The

variation of vertical forces a t t i r e s on the l e f t and right side of the

vehicle are plotted in Figure 16d for a maneuver which i s s l ight ly below

the rollover threshold. Wheel l i f t -o f f i s indicated in th is figure a t

points A and B.

The peak la teral acceleration level experienced by the t ractor

during a rollover was computed using the amplification factor ( A M P ) . This

type of calculation i s i l lustrated as follows for Configuration 6c.

MODIFIED 9AXLE DOUBLE , BKD 0067 , CONFIG *6C Figure 160

MODIFIED 9AXLE DOUBLE , BKD 0067 , CONFIG w6C

Figure 16b Ll n

MODIFIED 9AXLE DOUBLE , BKD 0067 , CONFIG *6C Figure 16c

LOO T I E ($%I

m 250 3.w 3.50 4.00

MODIFIED 9AXLE DOUBLE , BKD 0067 , CONFIG e6C

Figure I 6 d CI-

1 ) The peak la teral acceleration experienced by the t ractor -

was determined during the directional response simulation

of a lane-change maneuver with a steering input of one

degree ampl itude a t the front wheels ( for Configuration 6c

i t was found t o be 3.52 f t /sec2 or 0.11 9 ) .

2 ) The t ractor rollover threshold in g ' s was then computed as the product of the ampl i f icat ion factor (AMP) and the peak

g level determined in (1) ( for th is configuration AMP = 4.2;

therefore, the rollover threshold i s 4.2 x 0.11 or 0.46 g ) .

5.3.3 Summary of Rollover Threshold Level. The rollover threshold

levels are summarized in Figure 1 7 in a bar chart format, This-figure

shows the maximum lateral acceleration levels experienced by the t rac tor

during rollover in steady turning and 1 ane-change maneuvers.

For the vehicles analyzed, the rollover threshold levels during

steady turns were found t o range from 0.370 g for Configuration 6b (ninc-

axle double, BKD0067, with standard spring spacing and single t i r e s ) t o

0.4639 for Configuration 2b (six-axle tractor-semi t r a i lee , with a i r

suspension on a1 1 three t r a i l e r axles) . Tractor-semitrailers equipped with

ai r suspensions were found t o exhi b i t higher rol lover threshold levels

than the vehicles equipped with conventional springs. This, can be

attributed primarily t o the absence of backlash in the a i r suspensions.

As can be seen in Figure 1 7 , the rollover thresholds in lane-change-

type maneuvers were found t o vary over a wider range. For a l l of the

tractor-semi t r a i l e r configurations and the two modified double tanker

configurations (Vb and VIc), the rollover threshold in the lane-change

maneuver was found t o be higher (ranging from 0.46 t o 0.64 g ) t h a n in

the steady turning maneuver. On the other hand, the t ractor-ful l t r a i l e r

( IV) and the conventional doubles (Va, VIa, and VIb) exhi b i t much lower

rollover threshold levels (ranging from 0.18 g t o 0.28 g ) in the lane-change-

type maneuver. The modification of the draw bar i s seen to resu l t

approximately in a twofold increase in the rollover threshold level (compare Va with Vb and VIb with VIc). These resu l t s indicate that vehicles in- corporati ng conventional do1 1 ies tend to have low rol lover thresholds in lane-change (obstacle-avoidance) maneuvers.

5.3.4 Influence of Suspension Backlash. The influence of suspension backlash on rollover threshold was studied for the pup t r a i l e r of the ni ne-ax1 e double, BKD0067 (.Configuration VIa) . The resul ts of these calculations are plotted i n Figure 18 w i t h the suspension backlash as the abscissa and the rollover threshold level of the pup t r a i l e r as t M .,. --

- -.,. ~ ordinate. - ~ * -

- Suspension backlash i s found t o decrease the rollover threshold for -

-. both steady turning and 1 ane-change maneuvers, For back1 ashqa l ues of u p -

to 1.5 in., the decrease i n the 1 ane-change roll over t h r e s h o e i s found t o be more rapid than the decrease in the steady turning rollover threshold. - - - - - - Elimination of a backlash of 1.5 in. i s seen to increase the -. Tane-change I . . . -

rollover threshold by 18% and the rollover threshold in a steady turn by -

REFERENCES

1. Ervin, R.D., e t a l . "Ad Hoc Study of Certain Safety-Related Aspects of Double-Bottom Tankers." Final Report prepared f o r Office of Highway Safety Planning, Michigan Department of S t a t e Poi i ce , Contract No. MPA-78-ODZA, Highway Safety Research I n s t i t u t e , University of Michigan, Report No. UM-HSRI-78-18, ky 7 , 1978.

APPENDIX A

PARAMETERS FOR DIRECTIONAL RESPONSE CALCULATIONS

L i s t e d i n t h i s appendix a re veh i c l e parameters which were used i n

t h e d i r e c t i o n a l response ca l cu la t i ons . The parameters f o r t h e 38 cases

(see Table 2 f o r the d e s c r i p t i o n o f each case) a r e l i s t e d i n Tables A.l

through A.3. Table A.4 de f ines the symbols used i n these tab les . The

co rne r i ng f o r c e and a1 i g n i n g torque data f o r a Fruehauf 10x20 and a Uni-

r o y a l F leetmaster 11~22 .5 t i r e a r e p l o t t e d i n F igures A.1 and A.2,

r e s p e c t i v e l y . The Fruehauf 10x20 t i r e data was used on a l l veh ic les

except BKD0067, where t he Un i roya l 11~22.5 data was used.

Table A . 1

Table A.2

Table A.2 (~ont.)

Table A.3

Table A.3 (Cont.)

Table A.4. Definition of Vehicle Parameters

Note: A double subscript notation has been used when referring - to the axles on the articulated vehicle t ra in. An axle w i t h subscript i j denotes the j t h axle on the i t h element of the train. For example, the third axle of the semi- t r a i l e r (the semitrailer i s the second element of the t ra in) i s referred t o as axle "23."

'i weight of the i th element of the t ra in ( Ibs)

I i yaw moment of ' inertia of the i th element of the t ra in ( I b*in*sec2)

' i j sum of the cornering s t i f fness of a1 1 t i r e s mounted on axle i j (lb/deg)

sum of aligning moments/unit s l i p angle of a l l the t i r e s N i j mounted on axle i j (ft .1 b/deg)

'si j longitudinal s t i f fness of one t i r e on axle i j ( I b )

distance of axle i j from the mass center of the i t h 'i j element ( in )

distance of t ractor f i f t h wheel from mass center of ''A t ractor ( in )

' 2 ~ distance of t ractor f i f t h wheel from mass center of semitrailer ( i n )

distance of pintle hook from mass center of semitrailer 'ZB ( i n )

'3~ distance of pintle hook from mass center of dolly ( i n )

'3' distance of dolly f i f t h wheel from mass center of dolly ( i n )

'4' distance of do1 ly f i f t h wheel from mass center of pup t r a i l e r ( i n )

Yi j spacing distance between the dual t i r e s on axle i j ( i n )

ALIGNING TORQUE (ft- lb /drg)

300 250 300 150 100 1 I I I I

I I 1 1 I L

OOL 009 00s OOb 00s 002 c 16an/e11 ec7rl-I J I I a C ~ * I I U T W I U ~ - . -

VERTICAL LOAD (Ib) Figure A.2

APPENDIX B

EIGENVALUES A T 50 MPH

Natural Frequency and Damping Rat ios of t h e E igenva l ues a t a Forward Speed o f 50 mph.

* R E A L R O O T S

APPENDIX C

LATERAL ACCELERATION FREQUENCY RESPONSE PLOTS

TRRCTOR-SEMI BKY-8499 FULLY LORDED . DRTRSET*l

TRACTOR-SEMI BKY-8499 EMPTY . DATRSET*2

TRACTOR-SEMI BKD-0065 COMP#l FULL DATASET#6

TRACTOR-SEMI BKD-0065 COMP*4 FULL DATASET*7

TRACTOR-SEMI BKO-0065 COMP#U FULL.AXLE LIFTED. ORTRSET*lO

FREQUENCY (RRD/SEC) TRRCTOR-SEMI BKY-9Y5O-1 FULLY LORDED DRTRSET#ll

rd 1

I I

I

I I

8 a'--

t

k I Y

3 - -

0

-3- m d m H

$3 - S T W 0 2 I-

9%. - 1

CL: L

- -

-

- -

m TMCTOR

3 a SEMI TIWILER I I I I 1 I 1 I I l l ,

I I I I I l l I I

I I I

I I I 1 1 8 1 1 I I I

2 3 4 S 6 7 8 9 2 3 4 5 8 7 8 9 2 3 4 5 8 7 8 9 1x10'1 1 1x10' 1x102

TRRCTOR-SEMI BKY-9450-1 COMP#l FULL DATASET*13

TRACTOR-SEMI BK%-9450-1 COMPa4 FULL DATASET#lU

TRUCK/FULL-TRRILER BLY-27111 EMPTY .DflTflSET#lG

DOUBLE TRNKER BLY-2985 EMPTY .DATFISET*20

FREQUENCY (RQD/SEC) MODIFIED DOUBLE BLY-2985 FULLY LOADED .ORTRSET#23

MODIFIED DOUBLE BLY-2985 SEMI EMPTY PUP LORDED .DATASET826

FREQUENCY (RAD/SEC) DOUBLE TANKER BKD-0067 FULLY LORDED DQTQSET#27

DOUBLE TANKER BKO-0067 .SEMI LOADED PUP EMPTY. OATASET*29

DOUBLE TANKER BKD 0067 , SEMI FULL PUP-COMP*3 FULL , DFITASET*31

FREQUENCY (RAD/SECI MODIFIED DOUBLE BKD-0067 FULLY LOQDED DATRSET#33

MODIFIED DOUBLE BKO-0067.SEMI EMPTY PUP LOADED.DATASET*36

FREOUENCY (RAD/SEC) MODIFIED DOUBLE BKD-0067,SEMI FULL PUP-COMPr3 FULLSDATASET#37

MODIFIED DOUBLE BKD-0067.SEMI FULL PUP-COMP#243 FULL,DATASET#38

APPENDIX D

LATERAL ACCELERATION TIME HISTORIES DURING 2-SECOND EMERGENCY LANE-CHANGE MANEUVERS

AT 5 0 MPH

TRRCTOR-SEMI BKY-8499 EMPTY , DATflSET#2

TRUCK/ FULL-TRAILER BLY-27 1U , EMPTY , DATASET* 16

DOUBLE-TANKER-BLY-2985, SEMI T R ~ I LER~PUP EMPTY DATASET*20

SAXLE-DOUBLE-BKO0067.SEMI EMPTY PUP LOROEO OATFISET*30 I

1.00 1.50 TIME (SEC)

9RXLE-DOUBLE-BKD0067.SEMI FULL PUP#243COMP FULL DATASET#32

MODIFIED-DOUBLE-BKD-0067.SEMI LOADED PUP EMPTY DRTASET*35

MODIFIED-DOUBLE-BHOO067,SEMI FULL Pupa3 COMP FULL DATflSET*37

APPENDIX E

SUSPENSION MODELS AND LISTING OF COMPUTER PROGRAM

This appendix describes the two suspension spring models which are

available as options in the roll model. Following the discussion of the spring models, the computer program used in the roll simulation is listed

at the end of this appendix.

The two suspension models which are available as options are:

1) A model which represents the suspension by 1 inear springs

with coulomb friction and dead zone:

As shown in Figure E.l, the suspension backlash is represented in

the model by the dead zone "DEL." Two spring rates, "KS" and "KST," are used to represent the suspension stiffness in compression and tension,

respectively .

Dead I/ I Zone

Deflection -

Figure E.l

Special problems are present in developing a digital simulation

of a system with coulomb friction. The classical representation of

coulomb friction is of the form shown in Figure E.2.

Figure E.2

This form of representation is unsatisfactory for digital simulation since this leads to the system chattering about any given equilibrium point. A solution to the chatter problem, which has been found to be very effective, is the representation of coulomb friction in the form shown in Figure E.3,

Figure E.3

A more complete discussion on suspension coulomb f r i c t ion and the

computation of the parameter "DELBM can be found in Reference [I ] . This model i s especially useful when only rough estimates of spring rates and f r i c t ion levels in the suspension springs are available,

2 ) A suspension model which i s made t o f i t measured spring

data:

Measurements of suspension spring character is t ics a t HSRI (see

Reference [2]) and a t the Fruehauf Corporation revealed the relationship between vertical force and deflection t o be of the form shown in Figure

E.4. --

Figure E.4

This figure indicates that a model which represents the suspension by linear springs and coulomb f r ic t ion cannot be made t o f i t the measured spring data very we1 1 . A better representation of the complex behavior of the suspension springs i s made possible in t h i s model by the use of two force versus deflection tabu1 a r functions which envelope the measured data. Typical upper and lower envelopes are shown in Figure E.5.

/ Upper Envelop T// Deflection 4 4

Lower / Envelop

Figure E.5

In the digi ta l simulation, the rate a t which the force reaches the upper (or the lower) envelope during compression [or tension) i s approxi- mated by the following equation.

Fi = suspension force a t the current simulation time

Fi-l = representative suspension force a t the l a s t simulation time step

6 = suspension deflection a t the current simulation time step

6 i-1 = suspension deflection a t the l a s t simulation time step

F = force corresponding t o the deflection, 6 i , of the upper envi envelope for compress'ion and lower envelope for tension. This i s computed from the tabular representation of the envelopes,

= an input parameter used for describing the rate a t which the suspension force within the envelope loop approaches the envelope. 8 , i s used for the compressive portion of the loop, B 2 for the tension portion of the loop.

/Tensile Portion of Loop

Def lect ion -

Figure E. 6

The two parameters, 8, and e2, are chosen so as to accurately repre-

sent the hysteresis loops exhibited by the suspension springs.

REFERENCES :

1 . Winkler, C . B . , e t a l . , " P r e d i c t i n g t h e B r a k i n g P e r f o r m a n c e o f Trucks and Tractor-Trai l e r s - Appendix '3. " Phase I I I Technical Report, Truck and Tractor-Trailer Braking and Hand1 ing Project, Highway Safety Research Ins t i tu te , University of Michigan, Report No. UM-HSRI-76-26-2, June 1976.

2. MacAdam, C.C. "Computer Simulation and Parameter Sensi t ivi ty Study of a Commercial Vehicle During Antiskid Braking." Sixth IAVSD-IUTAM Symposium on Dynamics of Vehicles on Roads and Tracks, Berl in , September 3-7, 1979.

LISTING OF COMPUTER PROGRAM

C C C THIS PROGRAM COMPUTES THE ROLL RESPONSE OF A VEHICLE C TO A LATERAL FORCE INPUT. C THE INPUT-OUTPUT DEVICES ARE :

C DEVICE # C 5 ........ FILE CONTAINING VEHICLE PARAMETERS C 6 em........ OUTPUT DEVICE # C 7 .......... FILE CONTAINING TIME HISTORY OF LATERAL FORCE C

EXTERNAL INPUT, OUTP, OUTP2, TABLE REAL 1 1 , 1 2 , K S ( 2 ) , K S T ( 2 ) , KT1, KT2, KT3, KT4, LB, KBS DIMENSION XXT(25) , YYT(25) , XXB(25) , YYB(25) DIMENSION Y ( 1 0 ) , DERY(10), PRMT(5), AUX(16,10) , HEAD(14) DATA COUL /'COUL1/ DATA EXPL /'EXPL1/ DATA RAMP / I R A M P I / DATA STEP /'STEZ1/ DATA SINE / 'S INE1/ DATA DISC / 'DISC1/ COMMON /EXPO/ NUMI, NUM2, XXT, W T , XXB, YYB, BETA1, BETA2 COMMON /ONE/ W1, W2, 1 1 , 1 2 , HI, H2, B2, A, HR COMMON /TWO/ KS, KST, KT1, KT2, KT3, KT4, DEL, T, S COMMON /THRGE/ CF, CV, CVT, CFST COMMON /FOUR/ DEL10, DEL20, DELB COMMON /FIVE/ XPRINT COMMON /EXC/ FCT, TIME, AMP, TMAX COMMON /DISCR/ N , F ( 5 0 0 ) , X ( 5 0 0 ) COMMON /SEARCH/ COAMP COMMON /BOOM/ HBOOM XNEG = -9999.00 NUMI= 0 FACTOR = 0.0 WRITE ( 6 , 4 1 0 ) READ ( 5 , 2 4 0 ) W1, W2, 1 1 , 1 2 , HI , H2, B2, A, HR WRITE ( 6 , 3 7 0 ) W1, W2, 1 1 , 1 2 , H I , H2, 8 2 , A , HR READ ( 5 , 4 0 0 ) FRICT I F (FRICT .EQ. EXPL) GO TO 10 READ ( 5 , 3 6 0 ) K S ( I ) , K S ( 2 ) r KT11 KT21 KT31 DELI T I S READ ( 5 , 3 6 0 ) CF, CV, CVT, CFSTt K S T ( 1 ) t K S T ( 2 ) t FACTOR WRITE ( 6 , 3 8 0 ) K S ( l ) , KS(211 K S T ( 1 ) t K S T ( 2 ) t KT11 KT21 KT31 KT41

IDEL, T, S WRITE ( 6 , 3 9 0 ) CF, CFST, CV, CVT, FACTOR GO TO 50

10 READ ( 5 , 3 6 0 ) KT1, KT21 KT31 KT41 TI S t CVT, CV READ ( 5 , 3 6 0 ) BETA1, BETA2

C DO 20 J = 1 , 2 5

XXT(J) = 0 .0 YYT(J) = 0.0 XXB(J) = 0.0

2 0 YYB(J) = 0.0

5 3 C 5 4 READ ( 5 , 3 1 0 ) NUMI, NUM2 5 5 C 5 6 DO 30 I = 1 , NUM1 5 7 3 0 READ ( 5 , 3 6 0 ) X X T ( I ) , yY'l'(I) 5 8 C 5 9 DO 40 I = 1 , NUM2 6 0 4 0 READ ( 5 , 3 6 0 ) X X B ( I ) , YYB(1) 6 1 C 6 2 WRITE ( 6 , 2 5 0 ) KT1, KT2, KT3, KT4, T, S , CVT 6 3 WRITE ( 6 , 2 6 0 ) 6 4 WRITE ( 6 , 2 7 0 ) BETA1, BETA2 6 5 WRITE ( 6 , 2 8 0 ) NUM1 6 6 WRITE ( 6 , 2 9 0 ) (XXT(I),YYT(I),I=l,NUM1) 6 7 WRITE ( 6 , 3 0 0 ) NUM2 6 8 WRITE ( 6 , 2 9 0 ) (XXB(I),YYB(I),I=~,NUM~) 6 9 50 CONTINUE 7 0 READ ( 5 , 3 6 0 ) HBOOM 7 1 6 0 READ ( 5 , 4 0 0 ) FCT 7 2 I F (FCT .EQ. DISC) GO TO 70 7 3 READ ( 5 , 3 6 0 ) TIME, AMP, TMAX 7 4 WRITE ( 6 , 4 2 0 ) FCT 7 5 I F (FCC .EQ. SINE) WRITE ( 6 , 3 2 0 ) TIME, AMP 7 6 I F (FCT .EQ. RAMP) WRITE ( 6 , 3 3 0 ) TIME, AMP 7 7 I F (FCT .EQe STEP) WRITE ( 6 , 3 4 0 ) AMP 7 8 I F (FCT .EQ. STEP .OR. FCT .EQ. SINE) WRITE ( 6 , 3 5 0 ) TMAX 7 9 I F (FCT .EQ. RAMP) WRITE ( 6 , 3 5 0 ) TMAX 8 0 GO TO 140 8 1 70 CONTINUE 8 2 C 8 3 DO 8 0 JJ = 1 , 200 8 4 8 0 F ( J J ) = 0. 8 5 C 8 6 c ......................................

J = 0 READ ( 7 , 4 3 0 ) HEAD

9 0 J = J + l READ ( 7 , 4 4 0 ) X ( J ) , F ( J ) I F ( X ( J ) .LT. 0 . 0 ) GO TO 100 GO TO 9 0

100 F ( J ) = 0.0 X ( J ) = 0.0 N = J - 1 WRITE ( 6 , 4 6 0 )

140 CONTINUE READ ( 5 , 4 7 0 ) PRMT(2), PRMT(3), PRMT(4) READ ( 5 , 3 6 0 ) XPRINT

C COMPUTE THE COULOMB FRICTION BREAK POINTS C

DELB = PRMT(3) * CF * ( ( 7 7 3 . 0 * ( W l + W 2 ) / ( W 1 * ~ 2 ) ) + ( 2 , 0 * S * S * ( I 1 + 1 1 2 ) / ( 1 1 * 1 2 ) 1 )

DELB = DELB * FACTOR C

1 50 CONTINUE I F (FRICT .EQ. EXPL) GO TO 160 DELI0 = W1 / (KS(1) + K S ( 2 ) ) GO TO 190

C 160 DO 170 I = 1 , NUM1

I F ((W1/2.0) .LT. YYT(1)) GO TO 180 170 CONTINUE

C 180 DELI0 = XXT(1 - 1 ) t ( ( X X T ( 1 ) - XXT(I - l ) ) / ( Y Y T ( I ) - YYT(1 - 1 ) ) )

1 * (W1/2.0 - YYT(1 - 1 ) ) 190 CONTINUE

DEL20 = (W1 + W2) / (KT1 t KT2 + KT3 + KT4) C

DO 200 J = 1 , 10 Y ( J ) = 0 . DERY(J) = 0 .

200 CONTINUE C

Y ( 9 ) = HI - H2 - HR DERY(7) = 1. / 2. DERY(9) 1 . / 2 . P R m ( 1 ) = 0 . READ ( 5 , 4 7 0 ) COAMP I F (COAMP .LE. 0 . ) GO TO 230 WRITE ( 6 , 4 8 0 ) HEAD, C O W CALL OUTPUT CALL HPCG(PRMT, Y , DERY, 1 0 , IHLF, INPUT, OUTP2, AUX) I F (PRMT(5) .NE. 0 . ) GO TO 230 I F (PRMT(5) .EQ. 0 . ) WRITE ( 6 , 5 0 0 ) XNEG I F (PRMT(5) *EQ. 0 . ) GO TO 150 I F (IHLF .GE. 1 1 ) WRITE ( 6 , 5 1 0 ) IHLF

230 STOP 240 FORMAT ( /9F10 .3) 250 FORMAT (1H , 'SPRING RATE OF TIRE SPRING l ' , 11X, I = ' , F 1 0 . 2 ,

1 'LB/IN1, /, 6X, 'SPRING RATE OF TIRE SPRING 2 ' , 11X, ' = I ,

2 F10 .2 , 'LB/IN1, /, 6X, 'SPRING RATE OF TIRE SPRING 3 ' , 11X, 3 I = I , F10.2, 'LB/IN1, /, 6X, 'SPRING RATE OF TIRE SPRING 4 ' ,

4 l l X , I = ' , F10.2, 'LB/IN1/ lH , 5X, 'HOR DISTANCE FROM', 5 ' UNSPRUNG MASS CG TO INNER TIRE=' , F 1 0 . 2 , ' I N . ' , /, 6X, 6 'HOR DIST FROM SPRUNG MASS CG TO SUSPENSION = I , F 1 0 . 2 , 7 ' I N . ' , /1H , 'TIRE VISCOUS DAMPING = ' , F10 .2 , 8 'LB .SEC/IN1 )

260 FORMAT (1H , 'SUSPENSION SPRINGS MODELLED A S A TABLE ENVELOPEf/ 1 1H , 4 7 ( 1 H * ) )

270 FORMAT (1H , 'EXPONENT FOR UPPER ENVELOPE = ' , F10.4/1H , 1 'EXPONENT FOR LOWER ENVELOPE = ' , F 1 0 . 2 )

280 FORMAT (IHO, ' # OF DATA POINTS FOR UPPER ENVELOPE = ' , 1 2 ) 290 FORMAT (1H , F10 .2 , 2X, F10 .2 )

3 0 0 FORMAT ( 1H0, ' # OF DATA POINTS FOR LOWER* ENVELOPE = ' , I 2 ) 3 1 0 FORMAT ( 2 1 2 ) 3 2 0 FORMAT ( I X , 'TIME OF ONE PERIOD = I , F10 .2 , 'SEC. ' , /, 2X,

1 'AMPLITUDE =', F10 .2 , ' L B S ' ) 3 3 0 FORMAT ( ' O ' , lX, 'AT TIME =' , F10 .2 , 'SEC ' , /, 2X, 'FORCE = I ,

1 F 1 0 . 2 , ' L B S ' ) 3 4 0 FORMAT ( ' O ' , lX, 'FORCE=', F10.2 , ' L B S ' ) 3 5 0 FORMAT ( ' O ' , 1X, 'INPUT CONTINUES FOR', F 8 . 2 ,

1 'SEC., THEN VANISHES') 3 6 0 FORMAT ( 1 0 F 1 0 . 3 ) 3 7 0 FORMAT (6X, 'SPRUNG WEIGHT = ' , F10.2, ' LB. ' , /

1 , 6X, 'UNSPRUNG WEIGHT = I I F10 .2 , ' L B . ' , 2 /, 6X, 'POLAR M . I . OF SPRUNG MASS = ' , F10 .2 , 3 ' LB-SEC**2-IN', /, 6X1 'POLAR M . 1 . OF UNSPRUNG MASS - - 4 ' , F 1 0 . 2 , ' LB-SEC**2', ' - I N 1 , /, 6X, 'STATIC HEIGHT OF SPRUNG MAS 5 s CG = I l F10 .2 , ' I N . ABOVE GROUND', /, 6X1 'STATIC HEIGHT OF UNS 6PRUNG MASS CG=' , F10 .2 , ' I N . ABOVE GROUND', /, ' O ' , 5X, 7 'DISTANCE FROM SPRUNG', ' MASS CG TO I T S BOTTOM = ' , F 1 0 . 2 , 8 ' I N . ' , /, ' O ' , 5X, 'SPACING ', 'BETWEEN TIRES 9 =', F10 .2 , ' I N . ' , ' O ' , SX, 'DISTANCE FROM SPRUNG MASS CG TO CE *NTER OF ROTATION=', F10 .2 , ' I N . ' )

3 8 0 FORMAT ( ' O ' , SX, 'SPRING RATE OF RIGHT SUSPENSION SPRING I N ', 1 'COMPRESSION =' , F10 .2 , ' L B / I N V , /, 6X, 2 'SPRING RATE OF LEFT SUSPENSION SPRING I N COMPRSSION', 3 ' = I , F 1 0 . 2 , ' LB/IN1/ lH , 6X, 'SPRING RATE OF RIGHT SUSPEN 4SION SPRING I N TENSION = ' , F10 a2 , ' LB/IN1 /1H , 6X, 5 'SPRING RATE OF LEFT SUSPENSION ' , 'SPRIG IN ', 6 ' TENSION = ' , F 1 0 . 2 , ' LB/IN1, /, 6X, 'SPRING RATE OF TIRE 7SPRING l ' , 11X, ' = I , F 1 0 . 2 , 'LB/IN1, /, 6X, 'SPRING RATE OF TIRE S 8PRING 2 ' , 11X, I = ' , F 1 0 . 2 , 'LB/ IN1, /, 6X, 'SPRING RATE OF TIRE SP 9RING 3 ' , 11X, '= ' , F10.2, 'LB/ IN8, /, 6XI 'SPRING RATE OF TIRE SPR *ING 4 ' , 11X, I= ' , F10 .2 , 'LB/ IN1, /, ' O ' , SX, 'BACKLASH I N ' , 1 ' SUSPENSION SPRING=', F1O.2, ' I N . ' , /, ' O ' , 5X, 2 'HOR DIST. FROM', ' UNSPRUNG MASS CG TO INNER TIRE= ' , 3 F10 .2 , ' I N . ' , /, 6 x 1 'HOR DIST FROM SPRUNG MASS CG TO SUSP 4ENSION =', F 1 0 . 2 , ' I N . ' )

3 9 0 FORMAT ( ' o ' , 2X, 'COULOMB FRICTION IN EACH SPRING (COMPRESSION) = ' 1 F l 0 . 2 , ' L B t / l H , 2X, 'COULOMB FRICTION IN EACH SPRING ( T 2ENSION) =' , F 1 0 . 2 , ' L B ' , /, 6X, 'COEFFICIENT OF VISCOUS DAMPING I 3N' , ' EACH SUSPENSION =' , F10 .4 , 'LB-SEC/IN1, /, 6X, 4 f~~~~~~~~~ OF TIRE VISCOUS^, I DAMPING - - I 5 F10.4/1H , 'COULOMB FRICTION BREAK POINT FACTOR FOR SUSP. S 6PRINGS = I , F10 .2)

4 0 0 FORMAT ( A 4 ) 410 FORMAT ( ' 1 ' )

20 FORMAT ( ' O ' , 1X, 'INPUT FUNCTION I S ' , 2X, A 4 ) 4 3 0 FORMAT (14A4) 4 4 0 FORMAT ( 2 E 1 1 . 4 ) 450 FORMAT ( 2 F 1 5 . 5 ) 460 FORMAT ( ' O ' , 5X, 'DISCRETE INPUT') 470 FORMAT ( 3 F 1 5 . 9 ) 480 FORMAT ( T I , 14A4, 4X, 'AMPLIFIACATION=', F6.2)

490 FORMAT (TI, 'DATA FROM:', A4, ' INPUT', 5X, 'AMPLIFICATION = I ,

1 F10.2) 500 FORMAT (TI, E11.4) 510 FORMAT (2X, 'IHLF=', 13) 520 FORMAT ('O', 'INPUT RAMP RISES FOR', F8.2, ' SEC., THEN',

1 REMAINS CONSTANT') END SUBROUTINE INPUT(Xl Yl DERY) DIMENSION XXT(25), YYT(25), XXB(25), YYB(25) DIMENSION Y(10), AA(5,5), AT(5), F(5), DERY(10), FLAST(2),

1 DUST( 2 DIMENSION SV1( 15), DL(2), DDEL(2), FORC(2), FORC1 (2) COMMON /EXPO/ NUMI, NUM2, XXT, YYT, XXB, YYB, BETA1, BETA2 COMMON /ONE/ W1, W2, 11, 12, HI, H2, B2, A, HR COMMON /TWO/ KS, KST, KT1, KT2, KT3, KT4, DELI T I S COMMON /THREE/ CF, CV, CVT, CFST COMMON /FOUR/ DEL10, DEL20, DELB COMMON /SIX/ F11, F22, F33, F44, F55, F661 FYI DL REAL 11, 12, KS(2)l KST(2), KT1, KT2, KT31 KT41 LBl KBS REAL MIl M2 MI = W1 / 386. M2 = W2 / 386. N = FEXT(X) B = B2 F1 = 0. F2 = 0. F31 = 0. F32 = 0. F41 = 0. F42 = 0. FD1 = 0. FD2 = 0.

c------------------------------------------

C 20 DL(1) = HI - B - H2 + DELI0 - (Y(9) + (HR - B)*COS108 - S*SIN108)

DL(2) = HI - B - H2 + DELI0 - (Y(9) + (HR - B)*COS108 + S*SIN108) DEL31 = -(T + A) * SIN8 - Y(7) + DEL20

DEL32 = -T * SIN8 - Y(7) + DEL20 DEL41 = T * SIN8 - Y(7) + DEL20 DEL42 = (T + A) * SIN8 - Y(7) + DEL20 DDEL(1) = -Y(4) + (HR - E) * (Y(5) - Y(3)) * SIN108 + S * (Y(5) - lY(3)) * COS108 DDEL(2) = -Y(4) + (HR - B) * (Y(5) - Y(3)) * SIN108 - S * (Y(5) - lY(3)) * COS108 IF (NUMI .NE. 0) GO TO 100

c----------------------------------- C SPRING FFSCTION -- COULOMB

DO 90 I = 1, 2 IF (DL(1) .GT. 0.0) GO TO 30 IF (DL(1) .LT. (-DELI) GO TO 60 FORC(1) = 0.0 GO TO 90

30 IF (ABS(DDEL(1)) .GT. DELB) GO TO 40 FORC(1) = DDEL(1) * CV + (DDEL(I)*CF/DELE) + DL(1) * KS(1) GO TO 90

40 IF (DDEL(1) .GT. 0.0) GOT050 FORC(1) = DDEL(1) * CV - CF + DL(1) * KS(1) GO TO 90

50 FORC(1) = DDEL(1) * CV + CF + DL(1) * KS(1) GO TO 90

60 IF (ABS(DDEL(1)) .GT. DELB) GO TO 70 FORC(1) = DDEL(1) * CV + (DDEL(I)*CFST/DELB) +KST(I) (DL(1) +

1 DEL) GO TO 90

70 IF (DDEL(1) .GT. 0.0) GOT0 80 FORC(1) = DDEL(1) * CV - CF + (DL(1) + DEL) * KST(1) GO TO 90

80 FORC(1) = DDEL(1) * CV + CF + (DL(1) + DEL) * KST(1) 90 CONTINUE

C GO TO 140

C SPRING FRICTION EXPONENTIAL -- CURVE FIT c------------------------------------ 100 CONTINUE

C DO 130 J = 1, 2 IF (X .EQ. 0.0) DLAST(J) = DL(J) IF (X .EQ. 0.0) FLAST(J) = W1 / 2.0

* IF (DDEL(J) .GT. 0.0) GO TO 110

ZZ = DL(J) ZZL = DLAST(J) CALL TABLE(1, NUM2, XXB, YYB, ZZ, FSENV) CALL TABLE(1, NUM2, XXB, YYB, ZZL, FSENVL) EETA = BETA2 GO TO 120

110 ZZ = DL(J) ZZL = DLAST(J)

CALL TABLE(1, NUMI, XXT, YYT, ZZL, FSENVL) CALL TABLE(1, NUMI, XXT, YYT, ZZ, FSENV) BETA = BETA1

120 DELL = ABS(ZZ - DLAST(J)) FORCl(J) = (FLAST(J) - FSENVL) * EXP(-DELL/BETA) + FSENV FLAST(J) = FORCl(J) FORC(J)=FORCl(J)+CV*DDEL(J)

130 DLAST(J) = ZZ C

140 CONTINUE C c----------------------------------------------- C

DDEL31 = -(T + A) * Y(3) * COS8 - Y(2) DDEL32 = -T * Y(3) * COS8 - Y(2) DDEL41 = T * Y(3) * COS8 - Y(2) DDEL42 = (T + A) * Y(3) * COS8 - Y(2) FD31 = 0. FD32 = 0. FD41 = 0. FD42 = 0. IF (DEL31 .GT. 0.) FD31 = CVT * DDEL31 IF (DEL32 .GT. 0.) FD32 = CVT * DDEL32 IF (DEL41 .GT. 0.) FD41 = CVT * DDEL41 IF (DEL42 .GT. 0.) FD42 = CVT * DDEL42 IF (DEL31 .GT, 0.) F31 = DEL31 * KT1 IF (DEL32 .GT. 0.). F32 = KT2 * DEL32 IF (DEL41 .GT. 0.) F41 = KT3 * DEL41 IF (DEL42 .GT. 0.) F42 = KT4 * DEL42 AA(1,l) = M2 + MI AA(2,2) = (MI + M2) AA(1,3) = -M1 * Y ( 9 ) * COS8 AA(2,3) = -MI * Y(9) * SIN8 AA(3,3) = I2 + MI * Y(9) ** 2 AA(1,4) = -M1 * SIN8 AA(2,4) = M1 * COS8 AA(4,4) = MI AA( 1,5) = -MI * HR * COS10 AA(2,5) = -MI * HR * SIN10 AA(3,S) = MI * HR * Y(9) * COS108 AA(4,5) = MI * HR SIN(Y(8) - Y(10)) AA(5,5) = I1 + M1 * HR ** 2 F11 = FORC(1) F22 = FORC(2) F33 = F31 + FD31 F44 = F32 + FD32 F55 = F41 + FD41 F66 = F42 + FD42 F(1) = -MI HR * Y(5) * * 2 * SIN10 + 2. * Y(4) * Y(3) * MI * 1COS8 - MI * Y(9) * Y(3) ** 2 * SIN8 + FY F(2) = 2. * MI * Y(4) Y(3) * SIN8 + MI * Y(9) * Y(3) ** 2 * 1COS8 + MI * HR Y(5) * * 2 * COS10 - W1 - W2 + F33 + F44 + F55

2F66 F(3) = -2. * M1 * Y(9) * Y(3) * Y(4) + MI * HR * Y(9) * Y(5) * * 2 l* SIN108 + W1 * Y(9) SIN8 t FY * (H2 + Y(7)) + (F11 - F22) * S + 2 (F33*(T + A) + F44*T - F55*T - F66*(T + A)) * COS8 F(4) = MI * HR * Y(5) ** 2 * COS108 - W1 * COS8 + F11 + F22 + M1 *

1 Y(3) ** 2 * Y(9) F(5) = - 2 . * M1 * HR * Y(4) * Y(3) * COS(Y(8) - Y(10)) - HR * Y(9)

1 * MI * Y(3) ** 2 * SIN108 + W1 * HR * SIN10 - (F11 + F22) * (HR - 2 B) SIN108 - (F11 - F22) * S * COS108

C DO T50 I = 1, 5

C DO 150 J = 1, I

150 AA(1,J) = AA(J,I) C

160 CALL SIMQ(AA, F, 5, IER) IF (IER .NE. 0) GO TO 190

C DO 170 I = 1, 5

170 DERY(1) = F(1) C

DERY(6) = Y(1) DERY(7) = Y(2)

180 DERY(8) = Y(3) DERY(9) = Y(4) DERY(10) = Y(5) GO TO 200

190 WRITE (6,210) 2 00 RETURN 210 FORMAT (5X, I * * * * * MATRIX IS NOT POSITIVE DEFINITE * * * * * I 1

END SUBROUTINE OUTPUT COMMON /ONE/ W1, W2, 11, 12, HI, H2, B2, A, HR COMMON /TWO/ KS, KST, KTI, KT2, KT3, KT4, DEL, TI S COMMON /FOUR/ DEL10, DEL20, DELB COMMON /SIX/ F11, F22, F33, F44, F55, F66, FY, DL DIMENSION Y(10), DERY(10), AUX(16,10), PRMT(S), DL(2) COMMON /FIVE/ XPRINT COMMON /SEARCH/ COAMP COMMON /BOOM/ HBOOM REAL 11, 12, LB, KBS REFL KS(2), KST(2), KT1, KT2, KT3, KT4 I = 0

C C SET ROLL LIMIT = OUTRIGGER TOUCHING GROUND C

OUTL = SQRT((118.5)**2 + (HI - HBOOM)**2) THET = ATAN(118.5/(Hl - HBOOM)) RETURN ENTRY OUTP(X,Y,DERY,IHLF,NDIM,PRMT) XP = I * XPRINT X2 = X / 2.

10 I F (ABS(XP - X ) .LE. PRMT(3)) GO TO 20 RETURN

20 CONTINUE WRITE ( 6 , 8 0 1 X

C DO 3 0 J = 1 , 10

3 0 WRITE ( 6 , 7 0 1 J , Y ( J ) C

I = I + l RETURN ENTRY OUTP2 ( X , Y , DERY , IHLF,NDIM ,PRMT ) XP = I * XPRINT HX = HI - H2 - HR

C C LATERAG ACCN OF SPRUNG MASS C

Y12DD = DERY(1) - Y ( 9 ) * DERY(3) * C O S ( Y ( 8 ) ) - 2 . Y ( 4 ) * Y ( 3 ) * l C O S ( Y ( 8 ) ) + Y ( 9 ) * Y ( 3 ) ** 2 * S I N ( Y ( 8 ) ) - DERY(4) * S I N ( Y ( 8 ) ) - 2HR * DERY(5) * C O S ( Y ( 1 0 ) ) + HR * Y ( 5 ) ** 2 * S I N ( Y ( 1 0 ) )

C C ROLL ANGLE OF SPRUNG MASS C

PHI12 = Y ( 1 0 ) * 180. / 3 .14115927 HY7 = H2 + Y ( 7 ) HY9 = H2 + Y ( 7 ) + Y ( 9 ) * C O S ( Y ( 8 ) ) + H R * C O S ( Y ( 1 0 ) ) PHI2 = Y ( 8 ) * 180. / 3 , 1 4 1 5 9 2 7

C C CALCULATE THE HEIGHT OF OUTRIGGER TIRE C

HGHT = HY9 - OUTL * COS(THET - Y ( 1 0 ) ) HGHT1 = HY9 - OUTL * COS(THET + Y ( 1 0 ) ) I F (HGHT .GT. 0 . .AND. HGHT1 .GT. 0 . ) GO TO 4 0 I F (HGHT .LE. 0 . .OR. HGHTI mLE. 0 . ) PRMT(5) = 1 . XNEG = -9999.0 WRITE ( 6 , 1 0 0 ) XNEG WRITE ( 6 , 1 1 0 ) C O W , FY, PHI12, HGHT, HGHTl, HY9, THET, OUTL

4 0 CONTINUE I F (ABS(XP - X ) .LE. PRMT(3)) GO TO 50 RETURN

50 CONTINUE I = I + 1 WRITE ( 6 , 1 2 0 ) X , Y12DD, D E R Y ( l ) , PHI12 , HY7, HY9, PHI2 , F 1 1 , F 2 2 ,

1F33 , F44 , F55, F66 , FY, D L ( l ) , D L ( 2 ) 6 0 RETURN 70 FORMAT (2X, ' Y ( ' , 1 2 , ' ) = ' , E 2 0 . 1 0 ) 8 0 FORMAT (2X, 'TIME =' , F 1 5 . 5 ) 90 FORMAT ( 2 E 2 0 . 1 0 )

100 FORMAT ( T I , E11.4) 110 FORMAT ( ' O f , 5X, 'CO-EFF. OF AMPLITUDE = I , F 1 5 . 5 , /, 6X, 'FY = I ,

1 F15 .5 , /, 6X, 'SPRUNG MASS ROLL ANGLE=', F15 .5 , ' DEG.', /, 2 6 X , 'HEIGHT OF OUTRIGGER TIRES AT THIS INSTANT=', F 1 0 . 5 , 3 ' , ' , F 1 0 . 5 , ' INCHES ABOVE GROUND', /, 6X,

4 'HEIGHT OF SPRUNG MASS CG ' , F 1 5 . 5 , ' INCHES ABOVE GROUND', 5 /, 6X, 'OUTRIGGER LOCATION', /, 15X, 'ANGLE FROM SPRUNG MAS 6 s AXIS( VERTICAL)=' , F 1 0 . 5 , ' DEG', /, 15X, 'TOTAL LENGTH=', 7 F 1 0 . 5 , ' INCHES' )

1 2 0 FORMAT ( T I , 1 6 E 1 1 . 4 ) END FUNCTION FEXT(T) COMMON /EXC/ FCT, TIME, AMP, TMAX COMMON /DISCR/ N, F ( 5 0 0 ) , X(5OO) COMMON /SEARCH/ COAMP DATA RAMP /'RAMP1/ DATA STEP / ' S T E P t / DATA SINE / ' S I N E 1 / DATA DISC / ' D I S C 1 / I F (FCT .EQ. D I S C ) GO TO 1 0 FEXT = 0 . I F (FCT .EQ. RAMP .AND. T .LE. TIME) FEXT = AMP / TIME * T I F (FCT .EQ. RAMP .AND. T .GT. TIME) FEXT = AMP I F (FCT .EQ. S INE .AND. T .LE. TMAX) FEXT = AMP * S I N ( 2 . * 3 .

11415927*T/TIME) I F (FCT .EQ. STEP .AND. T .LT. TMAX) FEXT = AMP FEXT = FEXT * C O N RETURN

1 0 CONTINUE C

DO 2 0 J = 1 , N I F ( T .LE. X ( J ) ) GO TO 3 0

2 0 CONTINUE C

3 0 I F ( J .EQ. 1 ) GO TO 4 0 FEXT = F ( J - 1 ) + ( F ( J ) - F ( J - 1 ) ) / ( X ( J ) - X ( J - 1 ) ) * ( T - X(

1 5 - 1 ) ) GO TO 5 0

4 0 FEXT = F ( 1 ) 5 0 CONTINUE

FEXT = COAMP * FEXT RETURN END SUBROUTINE TABLE(Mt N, XI Yt Z t Q ) DIMENSION X ( 1 ) , Y ( 1 )

C DO 1 0 I = M, N

I F ( Z .LE. X ( 1 ) ) GO TO 2 0 1 0 CONTINUE

C 2 0 I F ( Z *NE. X ( 1 ) ) GO TO 3 0

Q = Y ( I ) RETURN

3 0 I F ( I .EQ. M ) I = M + 1

Q = ( Y ( I ) * ( z - X ( I - 1 ) ) - Y ( I - l ) * ( Z - X ( I ) ) ) / ( X ( I ) - X ( I - 1 ) 1 )

RETURN END

APPENDIX F

ROLL PARAMETERS

Listed in this appendix are vehicle parameters which were used in

the rollover threshold calculations. These parameters describe the ro l l properties of each of the thirteen vehicle configurations which were analyzed. The thirteen configurations are defined in Table 1.

The mass, roll moment of iner t ia and dimensional properties of each vehicle configuration in the ful ly loaded condition are given in Table F.1. The symbols used in the table are defined in Table F.2. The parameters for the tractor-semi t r a i 1 ers (configurations Ia through I I I ) and the modified double tankers-which are equipped with rigidized pint le hooks (configurations Vb and VIc) were computed by assuming the vehicles to be completely rigid in ro l l . The ent i re vehicle was, therefore, represented in the model by a single sprung mass and a single unsprung mass. Parameters which correspond t o such a composite vehicle representation are l i s ted in Table F.l for these nine configurations.

In the case of the truck-full t r a i l e r (configuration IV) and the double tankers equipped with a conventional pintle hook (Va, VIa, and V I b ) the ful l t r a i l e r i s not coupled in roll to the res t of the vehicle. Moreover, the full t r a i l e r i s the one that i s most susceptible t o a ro l l -

over. Hence, calculations of rollover threshold were made for these vehicles using parameters which describe the ful l t r a i l e r alone.

The measured force deflection characteristics of the three, multileaf suspension springs, UCD-9637, UCD-0511 and UXB0201, are portrayed i n

Figures F . 1 , F .2 , and F.3, respectively. The coordinates chosen to define

the upper and lower envelopes of the force-defl ection characteristics are also tabulated in these figures. (The representation of the springs in

the roll model i s discussed in detail in Appendix E . )

Roll properties of the T-type a i r suspension used in calculations made for the tractor-semi t r a i 1 ers , BKY8499 and BKD0065, were a1 so measured by the Fruehauf Corporation. From the measurements, i t was found tha t ,

due t o the active nature of the pneumatic system, th is suspension did not

Table F.2

w e i g h t o f sprung mass ( l b )

we igh t o f unsprung mass ( l b )

h e i g h t o f sprung mass c.g. above ground ( i n )

h e i g h t of unsprung mass c.g. above ground ( i n )

r o l l moment o f i n e r t i a o f sprung mass about i t s c.g. ( i n e l b - s e c 2 )

r o l l moment o f i n e r t i a o f unsprung mass about i t s c.g.

( i n e l b - s e c 2 )

h e i g h t o f sprung mass c.g. above t h e r o l l a x i s ( 1 b e i n - s e c 2 )

v e r t i c a l d i s t a n c e between sprung mass c.g. and t h e s p r i n g

hanger ( i n )

dua l t i r e spac ings ( i n )

suspension s p r i n g ha1 f spac ings ( i n )

l a t e r a l d i s t a n c e between t h e i n n e r t i r e and t h e c e n t e r

l i n e o f t h e v e h i c l e ( i n )

v i scous damping i n t h e suspension spr ing--used o n l y f o r

a i r s p r i n g s ( l b * s e c / i n )

v e r t i c a l s t i f f n e s s o f t h e t i r e s ( I b / i n )

a parameter wh ich desc r ibes t h e r a t e a t wh ich t h e suspension

f o r c e approaches t h e upper enve lope (see Appendix E)

a parameter wh ich d e f i n e s t h e r a t e a t wh ich t h e suspension

f o r c e approaches t h e l o w e r envelope (see Appendix E)

v i scous damping i n t h e t i r e s ( l b * s e c / i n )

possess a hysteresis loop similar t o the ones shown in Figures F.1

through F.3. The roll s t i f fness was also found t o be sensit ive to the vertical load carried by the suspension. Hence, a se t of four measurements were made with vertical 1 oads of 0 , 10000, 15000 and 20000 1 bs. An average l inear roll s t i f fness ra te of 241005 inmlbsldeg was obtained from the measurements. The a i r suspension was therefore represented in

the model by 1 inear springs of s t i f fness 9563 1 b/in, spaced 38-in. apart.

Since shock absorbers were used on the T-type a i r suspensions, i t was decided to represent them in the rol l model by a l inear viscous damper. An estimate of 80 1 b*sec/in, for the equivalent viscous damping of the shock absorbers was obtained from ride measurements made a t HSRI (see Reference [ I ] ) on a tractor-semi t r a i 1 er equipped with an a i r suspension.

Since no data was available for the t ractor suspensions, the following assumptions were made:

1 ) The t ractor rear suspension was assumed to be a multileaf

suspension, similar t o the ones used on the t r a i l e r s . In the case of tractor-semi t r a i l e r s B K Y 8499 and BKD 0065, the t ractor rear suspension was assumed to be an F2 with four- leaf springs, UXB0201, for a1 1 configurations ( I a through IId) including the ones with a i r suspensions on the semitrailer.

2 ) Since the t rac tor front suspension i s usually very com-

pl iant , i t was assumed that the rol l s t i f fness contributed by the t ractor front suspension was negligible.

3) For configurations where a mixture of a i r and leaf spring type suspensions are used on the vehicle, a composite force-deflection envelope was constructed. Figure F.4 shows the spring character is t ics used for configuration I Ia.

4 ) A rol l center height of 27 in. was assumed for a l l suspensions.

REFERENCE

1 . Nisonger, R.L. and Ervin, R . D . "Measurement of Ride Vibrations on Semi t r a i l e r s Incorporating Different Suspensions ," Technical Memorandum, Highway Safety Research Ins t i tu te , University of Michigan, September 5 , 1979.

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TANK TRAILER STABILITY ANALYSIS